Question

$\int sin(e^x)e^x dx =$

• A. $cos(e^x)+C$
• B. $-cos(e^x)+C$
• C. $sin(e^x)+C$
• D. None of these

Answer 1

Answer: (B)

$-cos(e^x)+C$

Answer 2

To solve the integral $\int \sin(e^x)e^x dx$, we use the method of substitution.

Let $u = e^x$.
Then, differentiate $u$ with respect to $x$:
$\frac{du}{dx} = \frac{d}{dx}(e^x) = e^x$.
This implies $du = e^x dx$.

Now, substitute $u$ and $du$ into the integral:
$\int \sin(e^x)e^x dx = \int \sin(u) du$

The integral of $\sin(u)$ with respect to $u$ is $-\cos(u)$.
So, $\int \sin(u) du = -\cos(u) + C$, where $C$ is the constant of integration.

Finally, substitute back $u = e^x$:
$-\cos(e^x) + C$.